Some AFMs show the autopilot minimum use height diagrammatically and this might relate to specific operations, e.g. AWO > Cat 1
Flight tests to determine the MUH are usually made from an ILS approach; thus the MUH assumes an obstacle clearance zone proportional to the ILS (1:29 slope?) Unless stated in the AFM, the AP does not have to disengaged at MDA, but pilots must be aware of the possibility of reduced clearance from obstacles / ground in the event of failure. In addition NPA could involve descent rates higher than those for an ILS (or assumed by certification), then the obstacle clearance may be even less; thus an early disengagement (before MUH) would be prudent.
For precision like approaches, as the MDA is approached the decision is either to GA (use AP as required), or land – disengage AP. If it is judged necessary to continue with the AP, then perhaps the land decision was marginal and a GA should be flown.
For the FD, leaving the display on or turning it off may not matter. It should not be used to aid visual flight after a land decision because the guidance for a NPA might be invalid – not to a GS origin. However, leaving the FD on may make it more accessible in the event of a GA from low altitude.